Unknown standard deviation of apple masses from a lower-tail probability, a binomial sample and a conditional probability

The mass, \(X\) grams, of an apple from a certain orchard is normally distributed with mean \(180\) and variance \(\sigma^2\).

It is known that \(\mathrm{P}(X<171.6)=0.2420\).

1. Find \(\mathrm{P}(171.6<X<180)\).[2]
2. 1. Find the standard deviation, \(\sigma\), of \(X\).
   2. Hence, find the probability that an apple chosen at random from this orchard has a mass of more than \(200\) grams.[5]

Fifteen apples from the orchard are chosen at random. The random variable \(Y\) is the number of these apples with a mass of more than \(200\) grams.

3. Find \(\mathrm{E}(Y)\).[3]
4. Find the probability that exactly two of the fifteen apples have a mass of more than \(200\) grams.[2]

An apple chosen at random from the orchard is found to have a mass of less than \(200\) grams.

5. Find the probability that its mass is between \(171.6\) grams and \(180\) grams.[3]